84.33.8 problem 20.8

Internal problem ID [22325]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 20. Regular singular points and the method of Frobenius. Solved problems. Page 109
Problem number : 20.8
Date solved : Thursday, October 02, 2025 at 08:37:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (x^{2}+2 x \right ) y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 45

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+(x^2+2*x)*diff(y(x),x)-2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 x \left (1-\frac {1}{4} x +\frac {1}{20} x^{2}-\frac {1}{120} x^{3}+\frac {1}{840} x^{4}-\frac {1}{6720} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (12-12 x +6 x^{2}-2 x^{3}+\frac {1}{2} x^{4}-\frac {1}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 64

ode=x^2*D[y[x],{x,2}]+(x^2+2*x)*D[y[x],x]-2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^2}{24}+\frac {1}{x^2}-\frac {x}{6}-\frac {1}{x}+\frac {1}{2}\right )+c_2 \left (\frac {x^5}{840}-\frac {x^4}{120}+\frac {x^3}{20}-\frac {x^2}{4}+x\right ) \]

✓ Sympy. Time used: 0.309 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (x**2 + 2*x)*Derivative(y(x), x) - 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{840} - \frac {x^{3}}{120} + \frac {x^{2}}{20} - \frac {x}{4} + 1\right ) + \frac {C_{1} \left (\frac {x^{2}}{2} - x + 1\right )}{x^{2}} + O\left (x^{6}\right ) \]